E. Metody Numeryczne
E.2. Schemat Heuna
rozwi ˛azywane w pracy ma ogóln ˛a posta´c:
x0 = f (x, t) + ξ(t) (E.4)
gdzie ξ(t) jest członem przypadkowym. Zało˙zymy krok czasowy ∆t, oraz fakt, ˙ze w obr˛ebie tego kroku czasowego warto´s´c członu przypadkowego jest stała, lub inaczej, ˙ze ¯ξ ≡ Z t+∆t
t
ξ(t)dt. Rozwi ˛azanie tego typu równania nast˛epuje w dwóch krokach. Najpierw liczymy predyktor(P), czyli zwykły krok metody Eulera:
xPn+1= xn+ f (x, t)∆t + ¯ξ∆t (E.5) Nast˛epnie liczymy warto´s´c funkcji w kolejnym kroku czasowym(korektor) przy u˙zyciu predyktora:
xn+1 = xn+1
2(f (xPn+1, t + ∆t) + ¯ξ) + (f (xn, t) + ¯ξ) ∆t (E.6) Schemat jest poprawny dla równa´n skalarnych jak i wektorowych. Poniewa˙z jednak schemat Heuna nie zachowuje normy wektora, nale˙zy po ka˙zdym kroku go normalizowa´c.
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